Decomposable tensors in $\Lambda^2 {\mathbb{C^n}}$

Question

Let $V=C^n$. How can I describe all decomposable tensors in $\Lambda^2 {V}$ for $n=3$ and $n \ge 4$?

Answer 1

Via the Plücker relations. I'll let $e_1,\ldots,e_n$ be the standard basis of $\mathbb{R}^n$ and let $\omega=\sum_{i,j}a_{i,j}e_i\wedge e_j$ be a typical element of $\bigwedge^2\mathbb{R}^n$ where the sum is only over the $i$ and $j$ with $i<j$. Then $\omega$ is decomposable iff $$a_{i,j}a_{k,l}-a_{i,k}a_{j,l}+a_{i,l}a_{j,k}=0$$ for all $i$, $j$, $k$, $l$ with $i<j<k<l$.